Trajectory-Dissipation Learning-Rate Controller

(1) Integrate the controller around SGD or Adam, without changing the network or loss. Every control interval of $T=20$ to $100$ updates, save parameter increments, minibatch identities, gradient estimates, and optimizer noise. Construct $K$ reverse surrogates by replaying the same minibatches in reverse order, applying the optimizer schedule backward, and using reversed Gaussian noise when available. For Gaussian update noise with covariance $C_s$, estimate each path log-density from the observed increment $\delta_s=\theta_{s+1}-\theta_s$ using $\log P_s(\delta_s)=-\frac12(\delta_s-m_s)^\top C_s^{-1}(\delta_s-m_s)-\frac12\log\det(2\pi C_s)$, where $m_s$ is the optimizer-predicted mean update and diagonal $C_s$ is estimated from recent update residuals. (2) Pseudocode: collect $K$ forward windows; compute forward and reverse Gaussian log probabilities; set $\widehat d=(KT)^{-1}\sum(\log P_F-\log P_R)$; update $\eta\leftarrow\operatorname{clip}(\eta e^{-\kappa(\widehat d-d_{\max})})$; continue training. (3) The fluctuation-theorem identity and dissipation interpretation come from the paper; path probabilities, effective temperature, covariance, and $d_{\max}$ are estimated empirically. Use block-bootstrap confidence intervals and freeze the controller when uncertainty is high. (4) First test: train an MLP and 2-layer CNN on MNIST or CIFAR-10, comparing fixed learning rates, cosine decay, and this controller across the fixed-step divergence region. The quantitative prediction is that $\widehat d_t$ rises before loss explosion, and controlling it extends the maximum stable learning rate. With $K=32$ and $T=50$, require the detected dissipation-rise point to precede loss explosion by at least 5 updates and reproduce the baseline stability boundary within 20%.